If we assume \(U_t,U_b,U^{\prime }\)  are all much larger than the hopping \(t_t,t_b\) , we can have a Mott insulator at total filling \(n_t+n_b=1\) , as illustrated in Fig. REF (a). This is true in WSe\(_2\) -WS\(_2\) -WSe\(_2\)  system or twisted AB stacked WSe\(_2\)  bilayer[1]}. However, we will also consider the case that the top layer is less correlated and has a larger \(t_t\) . This is true in the moiré+monolayer system such as in the recent experiments[2]}, [3]}. In this case we expect that the Mott gap survives only when the density of the top layer is smaller than a critical value, as illustrated in Fig. REF . In this paper we mainly focus on the case with \(n_t=x, n_b=1-x\)  with small \(x\) , so there is a charge gap and the difference between the moiré+moiré and moiré+monolayer system does not matter too much.
